Finite Group Databasebeta

← #10131

Subgroups — #10131

Order 256.

249 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 249 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3236C211yesmaximal, characteristic, complemented
2#3208C211yesmaximal, characteristic
3C2×#490C211yesmaximal, characteristic
4#1555C211yesmaximal, characteristic
5#1531C211yesmaximal, characteristic, complemented
6#1470C211yesmaximal, characteristic, complemented
7#1302C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×#127C2211yescharacteristic
10C2×#125C2211yescharacteristic
11#501—21no—
12#492—21no—
13#491—21no—
14#490—22no—
15#490—22no—
16#486—21no—
17#484—21no—
18#483—21no—
19#479—22no—
20#460—22no—
21#450—22no—
22#432C2211yescharacteristic
23#432C2211yescharacteristic
24#431C2211yescharacteristic
25C2×#102—21no—
26C2×#102—21no—
27C2×#97C2211yescharacteristic
28#331—22no—
29#330—22no—
30#324—22no—
31C22×D4—21no—
32C22×D4—21no—
33C22×D4—21no—
34C2×SD16—24no—
35C22×C8—21no—
36C4⋊Q8D411yescharacteristic
37C4⋊Q8D411yescharacteristic
38#127—21no—
39#127—21no—
40#127—21no—
41#127D412yes—
42#125—21no—
43#125—21no—
44#125—21no—
45#125—21no—
46#125—21no—
47C22⋊Q8—21no—
48C22⋊Q8—21no—
49C4×Q8—21no—
50C4×Q8—21no—
51#117—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4D411yescharacteristic
55C2×C4⋊C4D411yescharacteristic
56C2×C42C2311yescharacteristic
57#107—22no—
58#106—22no—
59C4⋊C8—21no—
60C4⋊C8—21no—
61C4⋊C8D412yes—
62#103—22no—
63#103—22no—
64#102—24no—
65#102—24no—
66#102—24no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69C22⋊C8D412yes—
70#97—21no—
71#97—21no—
72#97—21no—
73#97D412yes—
74C4×C8—21no—
75C24—41no—
76C2×Q8—21no—
77C2×Q8—21no—
78C2×D4—22no—
79C2×D4—22no—
80C2×D4—22no—
81C2×D4—44no—
82C2×D4—41no—
83C2×D4—41no—
84C2×D4—41no—
85C2×D4—82no—
86C2×D4—82no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91C22×C4#4111yescharacteristic
92SD16—84no—
93C2×C8—22no—
94C2×C8—22no—
95C2×C8—22no—
96C2×C8—42no—
97C2×C8—41no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C22⋊C4—41no—
113C42—41no—
114C42C2×D411yescharacteristic
115C42C2×D411yescharacteristic
116C42C2×D411yescharacteristic
117C42C2×D411yescharacteristic
118C23—42no—
119C23—82no—
120C23—82no—
121C23—82no—
122C23#12111yescharacteristic
123Q8—42no—
124Q8—42no—
125D4—44no—
126D4—44no—
127D4—82no—
128D4—82no—
129D4—82no—
130D4—82no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—42no—
139C2×C4—81no—
140C2×C4#12011yescharacteristic
141C2×C4#12011yescharacteristic
142C2×C4#12011yescharacteristic
143C2×C4#12111yescharacteristic
144C2×C4#12111yescharacteristic
145C2×C4#12711yescharacteristic
146C8—42no—
147C8—42no—
148C8—42no—
149C8—81no—
150C22—21no—
151C22—21no—
152C22—82no—
153C22—82no—
154C22—82no—
155C22—164no—
156C22#39211yescharacteristic
157C22#47112yes—
158C4—21no—
159C4—21no—
160C4—21no—
161C4—21no—
162C4—21no—
163C4—81no—
164C4—81no—
165C4—81no—
166C4—81no—
167C4#45412yes—
168C2—22no—
169C2—162no—
170C2#200311yescharacteristic
171C2#200811yescharacteristic
172C2#201911yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC2×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3236C2semidirect
5. #1531C2semidirect
6. #1470C2semidirect
7. #1302C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#32362
#32082
C2×#4902
#15552
#15312
#14702
#13022